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Vibrations of spherical nuclei
I.D.4
[
Nuclear Physics 2 4 (1961) 1---17; @ North-Holland Publishi,zg, Col, Amsterdam
!
Not to be reproduced by photoprint or microfilm without written permission froth; the publisher
VIBRATIONS
OF SPHERICAL NUCLEI G. E. B R O W N
NORDITA,
Copenhagen,,
and J. A. E V A N S a n d D. J. T H O U L E S S
Mathematical Physics Deparlme~t, Universil,t ol Birmi~gham R e c e i v e d 18 N o v e m b e r 1960
A b s t r a c t : T h e p a r t i c l e - h o l e i n t e r a c t i o n is d i s c u s s e d as t h e m e c h a n i s m for p r ~ : : u c i n g vibra~tional s t a t e s in nuclei. T h e p r o c e d u r e is first i l l u s t r a t e d b y m e a n s of a s c h e m a t i c model, from w h i c h i t is s h o w n t h a t t h e u s u a l t y p e of s h e l l - m o d e l c a l c u l a t i o n m u s t b e extt:n(led ~,} i n c l u d e c,>r:el a t i o n s in t h e g r o u n d s t a t e of t h e n u c l e u s b e f c , e it c a n 1)e applied to t h e c a l c u l a t i o n of xl~>rat i o n a l s t a t e s . R e s u l t s of c a l c u l a t i o n s in ]--] c o u p l i n g , b u t w i t h z e r o - r a n g e forces, are gP.:en.
1. I n t r o d u c t i o n
Considerable experimental 1-5) and theoretical 5,6) evidence s~lggest~ !:he existence of octupo!e vibrations in a large number of nucM ?-~>s~..,~sm s'',~ spheri~cal equilibrium shape. In this note we wish to indicate, that the particle-t~o:(~ interaction produces a low-lying 3-- state possessing an extremely large octupole moment and thus provides a natural mechanism for such a state. The techniques will be outlined in general form, however, so as to be applicable to other cases. It has been suggested 7, s, 9) that the particle-hole interaction is the mechan:ism producing the dipole state (giant photonuclear resonance), and zero-range calculations showed it to be extremely effective both in tran:¢ferring dipole strength to the dipole state and in raising tI~e energy of the latter. In refs. a, 9) it was pointed out that the particle-hole interaction was generally repulsive in T = 1 states and attractive in T = 0 ones. If, therefore, a T = 1 state carrying most of the multipole moment goes up in energy, then one would expect a T = 0 one to come down. The simplest cases to consider are double closed.. shell nuclei; it was pointed out in ref. s) that the valence particles played relatively little part in the state, except in determining its width. "Ihe simplest states to consider are those that arise from liI~ear combinations of particles raised from one shell to the next, i.e., excitations of ho) in the harmonic osciUator model. Now, whereas the dipole T = 1 state describes motion of neutrons against protons and, therefore, introduces no problems of spurious centre-ofmass motion, the dipole T = 0 state represents dipole motions of neutrons and 1 April 1961
2
:
~. E. BROWN et al.
protons together and is, hence, almost completely spurious. In the Elliott and Flowers calculation m), for example, this state was removed and discarded before the secular matrix was diagonalized. The next states one might look at are magnetic quadrupole or electric octupole state.s ~lthesestates have odd parity). We prefer now to look at the latte~-, since they are easily reached by inelastic particle scattering, and there is considerable evidence for their existel~ce. In this note we shall work mainly with a schematic model similar to that of refs. ~,8), quoting later results of shell-model calculations carried out in zerorange approximation. From the work with the schematic model it will become clear that the usual type of shell-model calculation will have to be modified considerably before being applied to the calculation of vibrational states because of the large energy shifts :involved. In section 2 we shall outline the treatment by the schematic model. In section 3 we consider effects from correlations present in the ground-state of the nucleus, and in section 4 we will show the classical correspondence of our schematic model. in the sections 5 and 6 we give results of shell-model calculations for O 16 and Ca a° in zero-range approximation. 2. The S c h e m a t i c Model In ref. 8) it was pointed out that we could represent the spin and isotopic spin character of a T : 1 particle-hole wave function as
"['he particle-hole interaction consi:~ts of two types of terms as shown in fig. 1, direct terms D and exchange terms E. For a Wigner force the interaction in
tl D
E
Fig, 1, Graphs corresponding to tke direct terms D and the exchange terms E of the particle-hole interaction.
T == i states was D and in T : 4) states, D - - 4 E . Although the interaction is more likely to be something like a Serber mixture plus a small spin-exchange, this will not look much different from the Wigner force in zero-range approxima-
VIBRATIONS
OF
SPHERICAL
NUCLEI
3
tion which we can consider in formulating our schematic model and which seems to reproduce qualitative features. We see t h a t the interaction in T = 0 states is d o m i n a t e d by the exchange terms. With a zero-range Wigner force, D and E are equal, so that the interaction is - - 3 E . Going to finite range decreases D compared with E, since the exchange term contains only one m o m e n t of the potential; i.e., in an expan:don
V(rl--r~) = ~ VL(rl, r.~)PL(COS 0,2)
(1]
L
only the term with L = 3 is involved in E for an octupole stat~::. This is easy t(~ see since the particle and hole coupled to angul;~r m o m e n t u m J cai1 ~i,nnihi]ate at a point as shown in fig. 1 only through the /th term in the expansion (1). On the other hand, D depends on contributions from a large number of moments of the potential Vr., and going to finite range decreases the contribution * from large L. We shall consider mainly the exchange term, therefore. i
i
1 I !
I! Fig. 2. Particle-hole crez~tion by absorption of a }J-ray,
We note, now. that if the m o m e n t VL(r ~, r.~) were factorizable and proportional to r~r r 2L, t h e n the matrix element V . , jj, between states ii' and jj', which we assume to be d o m i n a t e d by the exchange term, would be proportioilat to D . , D j : , where D . , is the amplitude for absorption of an octup(:.le y-ray. leading to creation of a particle in state i and a hole in state i', as shown in fig. '2. Now, in fact, Vr(r x, re) is nothing like rl L rez"since V r vanishes rapidly as ttl:,~ difference Ir~--rlf increases beyond 10 -la cm or so. However. we arrive at essentially the same representation of V . , z' if we keep all factor:.; coming from angular integrals, but set all radial integrals entering into both V . . ~. and D . , equal (see ref. 7)). We shall, therefore, use V.',z" = mI ,,/r/-"D.,D~j.,
(2)
where '/: is a constant determining the strength of particle-hole interaction in the schematic model. The + sigh refers to T = 1 states, the -- sign to T ~= 0 ones, and ~ will in general have different m a g m t u d e s in tlle two cases. In t h e calculations we have carried out withoua approximating the radial integrals in this way, the calculated V~:.., have often differed quite a bit from those of the schematic model. However, it has generally been found that the V.,, jj. are large t This point has been especiMly stressed b y Ben Mottelson ~md Aage Bohr,
4
G, E, BRO'~?N et a l .
(and negati~ve for T = 0 states) between states carrying large amplitudes D,e and D~j, and that they are small between states having small transition amplitudes. This seems to be sufficient to guarantee that qualitative features will com:e out as in the schematic model. Once one makes the above assumption abot~t the V,~,,~. in the T = 1 dipole case, one obtains immediately the results of ref. 7), finding in the degenerate case (all e~ in the secular matrix of that reference equal) that the dipole state is
#D = X D.'9.'/v/~D~¢,
(3)
2 and that it is pushed up an amount ~ . , ¢rDi¢ in energy. In dealing with the octupole case, we wish to obtain a solution for the case where all e~ are not equal. Thinking of the situation using a simp!e-harmonic well, for example, one has octupole absorption to states of unperturbed energies ?/~ and 3/~. and it may be important to include the latter. We prefer now to change the notation of ref. 7), so that ~i is replaced by ei---~,, where e~ is the energy of the excited particle in the potential well repres(~nting tile average interaction of tile A-particles, an4 ~, is the initial energy of the particle. In other words, e~--e,, is the unperturbed energy of the particlehole pair ii'. The SchrSdinger equation in matri!x form now reads
=
X
(4)
it"
and using the approximation of the schematic model eq. (2) for V.,, w, this is easily solved to give
ND., ~f"' = t:. --e~te,, ,
(4.1)
where N is a normalization constanL
Tilde energy of any given eigenstate may be found by taking the expectation w~ke of tile Ham iltonian, In the dege,:,erate case (all e~--e e equal to a common wdu~ ,:) the result~:~ arc particularly ~imple, the amplitudes ~v,~,being
(4.3)
~p., = N ' D . , ,
with ---- { Z
,
(4.4)
E = ~--+~ X D~,,.
(4.5)
if'
and the energy of the vibrational state if"
VIBRATIONS OF SPHERICAL NUCLEI
5
A slight generalization of the degenerate model is necessary to deal with the octupole case, since the absorbtion can proceed to states of unperturbe4 energies ~,/go and ,~ 3/~(o. Our schematic model would predict in this case (1) ~ (11
t ~ D . , v.,
(3)
(3)
,
where the upper suffices refer to states of energy 1/1o)and 31i~o, respectively'. Simple calculations then show that the second term has quite an important effect on increasing the octupole transition probability to the collective state~ Another way of including the effect of higher transitions, as suggested by Ben Mottelson, would be to introduce an effective octupole charge. However, we believe that one should ultimately try to directly calculate the effect of the higher transitions. The experiments '~) demand a higher transition probability than that obtained by Elliott and Flowers ,o), but we believe that thi~ comes about from the effects of ground-state correlations, which, we shall now 4iscu~;:~, as well as from the 3//o) transitions. 3. E f f e c t s f r o m C o r r e l a t i o n s
The necessity of taking correlations in the ground state into account in plasr~s, oscillations of an electron gas has long be recognized (See, for example, Sawada et al. 13) or FerreI1 and Quinn la)). Their inclusion is necessary to reproduce the Bohm-Pines theory is) and to make correspondence with the classical theory.
a
b
Fig. 3, a. Absorption of a ?+-ray and subsequeilt processes, b. Interact:ion of two parLicles in tim ground-state of the nucleus, forming two particle-hole pairs, a n d subsequent -~rocesses.
The necessity of including them in nuclear vibrational states has also) been pointed out le, 17). We essentially rephrase the arguments of ref. ,7) in our schematic model. Since v e deal now with vibrational states, we consider the exchange term of fig. 1, which is of main importance here. In fig. 3 we illustrate two processes which are topologically similar. Fig. 3a represents the process considered earlier, whereas fig. 3b represents two particles interacting it~, the ground state of the nucleus, forming two particle-hole pairs, one of which is
6
G.E.
BRO'~VN et al.
subsequently annihilated in absorbtion of the 7-ray. Clearly, the most conveni:ent way of including processes of the type fig. 3b is to use F e y n m a n techniques and allow for propagation in a chain either forwards or backwards in time. To this end we write down the Gell-ManmLow wave function :ts) ~,(1, 2) -~- i f f d v s d ~ 6 S v ( 1 ,
5)Sv(2, 6)1"(5, 6)~0(5, 6)
(6)
in their notation. Setting t~ = t~ ~ t' and t 5 =-. t e := t, letting coordinate 5 refer to a particle and coordinate 6 to a hole for t <2 f , and choosing /'(5, 6) = 3(ts--t~)V(5, 6),
we find in going over to an angular-momentum t
7'"'(t ) =
(6.1) representation that
. f oo dt SF~(t , --t)SFe(t r --t)~ 2".,,jj.~,~,(t),
~ -oo
(6.2)
where
we(t) ~ (ii']~p(1, 2)) = z . , e -~E'', iOie-iq(t'-O, SFi(t'-'t) =
--i(1
0 ~e-i~'tt'-°
- - i ( i ---Oe)e i~e(t'-t),
Sw.
(t' - - t) =
i O v e ~''" ( r - o ,
(t < t'), (t > o Ct ),
(6.3)
(t < t'), (t > c),
and Z.' w~mld be just the same as the previous ~p.. if one carried out the integration over only t < t'. Carrying out the integration, we find
Z." --= --
ei__av_E
~j, V..,j,r Z# ,
(7)
wh~we 0i = 1 if state i is nnoccupied and 0~ --= 0 if it is occupied. That is, the [actor 0,.(1---0e) guarantees t h a t the state i refers to a particle, the state i' to a bo!~::. The first term in eq. (7) reduces to our previous result. Tile second term rep,~.:~ents the effect of the excitation propagating backwards (these two parts are c;dled A~ ~+~ and Ai(-~ by Ferrell and Quinn 14)). Thus, although we started out ,xith i htbelling a particle and i' a hole (this designation referring to tile ba~e vacuum, where one can unambigously talk about particles and holes because of the Sharp discontinuity in occupation number at the Fermi energy), this convention has become reversed in the backwards part. However, particles can be created only in states of energy greater t h a n the Fermi energy EF, and ,holes in states of energy less than E~. Thus, i' in the second case takes on precisely the same values as i in the first case. We can preserve the designation of particles by i and holes by i' if we interchange i and i' in the second term,
VIBRATIONS OF SPHERICAL NUCLEI
7
obtaining 1 , ZJ'i(, ~" { V " " w Z ~ : + V""~'i" ~r
e~--ei ' - E
(s)
1
%-- ~.~--E ~r ~ {Vri, ~, Z:J'+ V,,~,:.:Z:i},
where Z.' refers to the forward-going part of the excitation, and Z,'~ refers to the backward-going part. The normalization is such 14,21) (see, also, eq. (14)) that Y Iz,.:-
~ Iz,.,i 2 = 1.
ti'
ii'
(s.i)
The quantities V will be discussed for the general case later. We note ti~at if one includes only the exchange term of fig. 1, i.e., graphs of the type shown in fig. 3, then Vii"
JJ' -- --
[ : i i ' , J'j =
Ve~, j j" =
V .d. G j ' j :,
(8~Zl
which is a great simplification. In this case we can add the two equations (8), obtaining 2(e,-Ee)
z.. + z,., -
(~-~,.)~--E~y'..V.. ,,.(z.,
+ z,,~),
Of
showing that the secular equation describing only the forward going part can be modified rather trivially to give the squares of the eigenenergies E 2. In the ~chematic model one finds easily that Zii"
ND., ei--ee-- E ,
ND., Zi'~ . . . . . . . . . . . . . . . , e i _ e i _t_ E
(9.1)
where N is again the normalization contant to be determined from eq. (8.1), We see that in the vibrational case when E < e~--e~,, tile.' inclu!,don of the backwards-going terms enhances the absorption. Tile energy is given by the dispersion law * 1 =
~
2 ~¢:'D~,,(ei -- ei,) .............. -)-----:-~-, (<--~,) - - E
(9.2)
and it is seen that inclusion of the backwards terms lowers the energy still more. In fact, in the degenerate case (all t.,i--e,v equal to e) E = v'~2-2~;i; 2 -~, . %- ~. b~ . . . ii" * Tiffs r e s u l t h a s also b e e n o b t a i n e d b y M. B a r a a g e r a n d Bell MottelsotL
(9.a)
s
G, F.. BROW.~et
at.
The transition amplitude in the general case is given by -----2~ D . , ( Z . , + Ze,),
(9.4)
if phase factors are chosen so that De~ = D.,, and, in the degenerate case the transition probability is given by t~:] ~ = -~ .., D 'ii',
(9.5)
that is, it is enhanced by the inclusion of backwards-going graphs by the factor e[E. The transition probability to this one state then exhausts the energyweighted sum rule, the inclusion of the backwards-going graphs being necessary for thi~ J~'17). 4. C l a s s i c a l C o r r e s p o n d e n c e
When the energy of the vibrational state becomes small compared with the single-particle excitations, the formulae shoud go over to those of the adiabatic case, and expanding in powers of E/(ei--~e) we should regain the semiclassical description t. To make the correspondence more obvious, we go over to the notation ohi. = ei--er, to = E. Then, with expansion in powers of co/~o.,, eq, (9.2) becomes
This is to be compared with the adiabatic formula 19) K ~ + M ~ " == Ko~--3Io~2~ = 0,
(11)
where our formula (10) must be essentially K - - M ~ ~ = 0 since no collective coordinate occurs. Here K represents the restoring force, and M, the inertial parameter. In fact, to within a constant, the coefficient of in eq. (10) is just the inertial parameter of the cranking model 2o). This is easy to see. The prescription of the cranking model is that one should add to the Hamiltonian H = Z Ito(i) == ~ ( T , + Vj) J
(12)
J
describing the motion of particles in a spherical well V~, the term (~[I == ~ ~ YLO(Oi, ~o~)rS L
(12.1)
in describing L th order vibrations; this additional term effectively distorts the shape of the well. Then
l(ii'l ~ !o)!3 ii' t This was suggested b y Aage Bohr.
O.~tt,
(1'.,.2)
VIBRATIONS ('F SPHERICAL NUCLEI
9
where [0> is the uncorrelated ground, state. But 0
1 OJ i i,
(ii'] ~ YL°(O~, q~j)r:r~!O) oJ i i,
(12.3) ADii, Okii,
Substituting from eq. (12.3) back into (12.2) shows that the coefficient of oJ2 in eq. (10) is, to within trivial constants, equal to M. It is not :~urprising that our eq. (2) should lead us back to the cranking-model inertial parameter, for it would be exactly fulfilled if VL(r ~ , re) were equal to rt L r.~L and we kept only the exchange terms. In this case keeping only the L th term of the nucleon-nucleon force in the expansion (1) as a residual force between particles moving in a spherical well leads directly to the particles mo,~:mg independently in a deformed we]l. The cranking model treat:; just this case. It is also not surprising that one must include ground-state correlations to make the classical eorre. spondence, since this situation is known in the treatment of plasma oscillatio~,~s in a metal 13, i,). As the number of terms in the sum of eq. (9.2) increase, or as the stre~agth of the potential '#~"increases, E moves down until it becomes zero, and any further increase in the aove factors would cause it to become imaginary. This is connected with the collapse of the nucleus into nonspherical equilibrimn shape 2,). 5. R e a l i s t i c P o t e n t i a l s
The cMculations which are described in the next section were carried out by solving eq. (8), but it was not assumed that the potential has the ~dmple form needed to give the schema,qc model of section "3. Eq. (8) is similar to a SchrSdinger equation for a wave function with components Ziv and Z~'~, but there are some important differences. We give here a brief discussion of the general properties of eq. (8), since these properties must be understood to interpret the results of the calculations. Eq. (8) can be wri,~ten as a matrix equation
where X is a colmnn vector whose components are the numbers Z,', and Y is a column vector whose components are the numbers )~v~.The equation takes this simple form, with A and B Hermitian matrices, if corresponding components of X and Y are Z,, and 2?7, where a bar over an index denotes the tinmreversed state. The matrix B has elements equal to V,,, j,j, and the matrix A has diagonal elements equal to e ~ - - e e - l - V , , , , , , and off-diagonal elements
10
G.E.
m ~ o w N et aL
V,, is" These matrices are Hermitian because of the time-reversal invariance of the shell-model energies and of the potential. Eq. (13) shows that the energy levels are given by the eigenvalues of the nonhermitian matrix
and the peculiar properties of the solutions of this equation arise from the nonhermitian property. The first thing to be noticed about eq. (13) is that for every positive eigenvalue E there is an equal negative eigenvalue --E, since eq. (13) implies (~ ~ ) ( ~ )
-=--
( Y E _X).
( ~.~'
These negative solutions have no physical significance, but describe the same excited states as the positive solutions. If we have two eigenvectors
with eigenvalues Em and En, we get
(13.2) Unless Era* and E. are equal, we must have
X~*Xn--Ym~:Y,~ -----0,
(14)
and this replaces the usual orthogonality property of eigenvectors of a Hermitian matrix. This relation shows why the normalization of the eigenvectors given i~ eq. (8.1) is the natural one to choose in this problem. A ~ufficient condition for the eigenvalues of eq. (13) to be real is that the eig~mvalues of the Hermitian matrix
should M1 be positive. This condition implies that
(A AB) ~X,,~ = E,~(X,~'X,,--YJY,,)
(15)
must be real and greater than zero, so En must be real. If the attractive part of the interaction is too strong, this condition breaks down, and imaginary eigen-
VIBRATIONS OF SPHERICAL NUCLEI
t] ~
values, representing unstable modes of vibration, may appear. It can also be shown that, if this condition is satisfied, there is a complete set of cigenvectors of eq. (13). It follows that, for any X and Y,
gives an upper bound for the magnitude of the lowest eigenvalue of eq. (13) 21). This variational principle shows that we overestimate the excitation energy of the lowest excited state when, as in section 4, we consider only a restricted number of configurations. Energy-weighted sum rules are particularly simple in this theory. We denote by Q the column vector whose components are matrix elements Q,, of an operator, such as the octupole operator, between the occupied state i' and the unoccupied state i. The matrix element of this operator between the ground state of the system and the excited state of energy E,, in our theory is
,,,,,
(~u.tOi~0) = (X.*Y,,*) O . The energy-weighted sum of the squares of these matrix elements is
where the sum is taken over all positive energy eigenvectors. From the completeness of the set of eigenvectors, and from the symmetry between positive and negative energy eigenvectors, we can show that this equation gives
~', iL(.?f.tY~*)
E,,
= . g1( Q f - Q ) I" ,
=Qt(A--B)Q.
( ~'~
If eq. (8.2) is satisfied, the interaction drops out ot this expression, and the sum is the same as it would be in the unperturbed system. This is the case in either our schematic model or in our later calculations with zero-range forces. Consequently, neither of these is adequate to describe the enhancement of the energy-weighted sum rule through effects from exchange forces in tile T = 1 case. However, it can be shown 2~) that this enhancement is contained in the formalism if oae employs forces of finite range and demands self-consistency.
6. Calculations in ]~] Couplin~ The most important results of this work are calculations in which some ground state correlations have been included as described in the preceding sections. For completeness, however, we begin by reporting results of calculations in which such effects are not included. The procedure, in this case, is precisely the
G . E. B R O W N
t:2
el al.
same as that of the earlier paper on the dipole states *), and again the work is confined to the closed shell nuclei 01~ and Ca 4°. As before 8), we indicate the effect of a realistic exchange mixture by calculating the matrices for both an ordinary force and a Soper Mixture. if only 1 /ho transitions are considered, the three configurations p½-'dj, p!:~ld[ and p~-ld~ give rise to an octupole state in 0 ~. The eigenvalues, together with the squares of the perturbed octupole moments (denoted by ~ ) are given in table la. As expected, most of the octupole strength is resident in
TABLE
1
Calculated energies and octupole strengths in 0 ~ ]
0
Ordinary force
Soper mixture
80
8.8
67
16.0
11
16.2
19
22.9
9
21.6
14
7.7
0
i
100%
~o0%
.5 3,fi
367
74
7.7
134
58
l&8
13
12
16.0
24
22
22.6
11
14
21.5
16
20
391%
lO0%
174%
100%
w,
i
!
!
I
1
is proportiomal to the octupole transition amplitude (ph[Qs[0), where Q3 = ra Y3°(0, cp). The %%; are taken with respect to ~t~.D.. 2 and 51.igei~.D.,Z, where the sum is over unperturbed wave Lm~tions, and e., = e~--e(, the unperturbed energy. All transitions considered are between adjac,e1~t ~hetls, and the sums go only over such configurations.
the ]c~,:~'est level, the energy of which could be fitted to the experimental value siml~],y by adjusting the interaction strength Vo. However, we thought it more useful to retain the value of Vo which fitted the O le dipole states 8), The eigenvector obtained for this state agreed, almost exactly with that of Elliott and Fl(Jwers, and consequently leads to their estimate of the lifetime. Using a reasonable value for the range paramef.er b of the wavefunctions, they obtained a lifetime three time,~s too long. We shall show how this situation is improved by the introduction of ground state correlation effects. For Ca ~, the results, in table 2a, are very similar. In tiffs case, nine octupole
VIBRATIONS
OF
13
SPHERICAL N U C L E I
states can be formed from the configurations d -~ f, d -~ 2p, and 2~s-~ f. Although the value of b 3 was increased in the ratio 40/16, V o was unaltered t. Correlation effects were included for both nuclei by solving the matrix equation (13) numerically. In zero-range, the calclflation of the matrix B is TABLE
2
Calculated energies and octupole strer~gths in Ca 4° E
Ordinary force (MeV)
G
6.0 8.7 9.8 11.1 12.6 14.2 16.5 16,6 17,9
",9
2 s2
1.9 8.5 9.5 ll.O 12.6 14.1 16.5 16.6 17.9
~
%
Soper mixture
E~ 2 %
62 13 15 0
2
ii
3 1
1
2
i
100%
i
2i
E(MeV)
~2 %
6.3 9.2 10.2 11.3 12.6 14.8 16.1 16.5 17.2
43 7 35 0 2 4 7 0 1
lOO%
512 8 16 0 2 3 4 0 2
72 5 11 0 2 2 5 0 3
5.4 9.0 9.9 11.3 12.6 14.1 16.1 16.5 17.2
10o%
547%
E ~ 2 %,
109 13 39 0 2 5 8 o 1
45 9 29 0 2 5 9 0 1
177°;
See table 1 for definition of the symbols.
rather shnple. The phases of the u n p e r t u r b e d states can be chosen so that all the oct'apole m o m e n t s ~ are positive and invariant under the interchange i =,- i',
i' --~-i.
T h a t is to say D.,
== Di,
i .
It follows then, t h a t the part of the matrix element correspondang to zero total ,spin S --- 0, is invariant under the interchange of particle and hole q u a n t u m numbers. A simple argument shows that under these circumstances the S -- 1 t After these cMculations were completed, it was pointed out to us by J. Blomquist t h a t one should take the r.m.s, radius (fi)t to be proportional to A t and not b. This would involve a sub:stantial increase in Vo for (,~,,,0 b u t would not change our results quMitatively.
6. E. BROWN et al.
14
part always changes sign. Now the S = 1 terms in the interaction can arise only as a result of spin-orbit forces, so that if the latter are negelected, the relations (8.2) must be satisfied (in zero-range approximation). In the present calculations, however, spin-orbit forces are included, and therefore B differs frc,m the interaction part of A. Two important features of the results for 0 ~6 in table lb, and for Ca ~ in table 2b, are that for both types of force, the energies of the lowest states are lower than the corresponding ones in la and 2a, and that these lowest states have become very sensitive to the exchange mixt a r e . It can be shown quite generally that the lowest eigenvalue must be lower than the lowest eigenvalue of the matrix A. We now turn r.o a discussion of sum rules. Lane and Pendlebury 6) have considered the fractions of certain sum rules exhausted by the lowest octupole s t a t e s in various nuclei, including O a6 and Ca4°. First they used the sum of squares of matrix elements E D~.,. ii'
and find, for the lowest 3- state in O ~6 that ~
(17)
~ 0 . 6 4 ~ P . ~, . ii'
Now, the total sum contains contributions from both l h ~ and 3tUo configurations, so that = ~i'
D., q- E
i,',
ii"
where the upper suffices "1" and "3" refer to ltwj and 3hoJ, respectively. It turns ou~ for 0 ~ that D(a?-
ii" i b.nJ:, :~ e,
ii"
__
--
7
-3 ~ ii'
D(n~
ii" "
from (17)
~@2= 0.64(1
+~)Z D(1)°'=ii'2 . 1 3 if"
l.~ ~ ~'it'/)(1)2
(18)
if'
Retatior, (18) :is the more convenient form to use here because we have not explicitly irmluded any 3ti~,~effects in our calculations. If the results of table lb (for the 5oper Mixture) are recalculated with a force strength V0 which tits the ]cnvest energy to the experimental value of 6.13 MeV ¢, then c~2 rises from 134 % to 184 %. This latter figure is to be compared with the 213 % in (18). If one does not simply renormalize, but tries to make an estimate, of what elfect t h e 3ho~ transitions would have on 2 *, one finds that they would give This "renormalization" m a y be thought of as taking some account of the 3ho9 transitions, inch~sion of which wouM push the level down. This is in the spirit of using an effective charge for "/¢ay transitions.
V I B R A T I O N S OF S P H E R I C A L N U C L E I
15
rise to an enhancement of about 40 %, leading to ~2
___ 1 . 3 4 ×
× ~ D(1)~ir= 1 . 8 8 Y~ l) wa)~,
1.40
ii'
ii"
in agreement with results from renormalizing the interaction. It is interesting to consider the reason underlying our large ~2. In the normal shell model, in which ground state correlations are not included, the orthonormality of the eigenvectors implies that,
Z n ('>' .
=
--it'
(l:))
'
it"
where the index p goes over all perturbed stat(.. From this it is clear that. for the iowest level, < D it"
making the result (18) unobtainable. Of course, the situation w(mld be improved by including 3h~o transitions, but it turns out that these do not mix in sufficiently strongly. It is, in any case. clear that there is no possibility within the framework of the shell model with 1/i.w transitions of obtaining a correct value for the matrix element, and hence of the lifetime of the low 3- resonance ;n O ~(~. When ground state correlations are introduced, (19) is no longer sati~fied. and indeed there is no theoretical limit on the left hand side; it will b(c)m( infinite as the lowest eigenvalue drops to zero. In this case, pr )~Acd only that zero range forces are used (and, in fact, for finite-range forces if self-consistency is enforced ~.1)), X E v ' c> ~ . 2 = X E~ . ' -l-). a)~ " . P
ii'
and E., being the perturbed and unperturbed eigenvalues re+;pectively. In the present case, we obtained with E =: 6.13. the result ED
E ~ 2 = 0.65 ~ E. . . L) (1) ., '' .
(20)
ig"
Now ideally the total energy-weighted sum would be given by
X E w l )2. .
=
8 ~. L~-i v I J i ir~tl): , ,
ti"
t(
and (20) would become O. ~a E~2
--
8
X Ei,:D " 2. , iY
:
0.08 ~ E.., 1)"' _.,. ig'
Increasing this by 40 % as before, we would obtain E,~ 2 ~ O.II ~, ~F., t '~-,l) ~i f p , if'
(21)
16
G.E.
BROWN et al.
which, bearing in mind the uncertainties in this procedure, agrees quite well with the estimate of 0.18 of this sum rule given by Lane and Pendlebury e).
7. Summary In this and a preceding paper 8), the origin of vibrational states through the mixing of shell-model configurations by the particle-hole force has been discussed. The calculations carried out in these papers were only rough ones, and, in some cases, were inconsistent. For example, the enhancement of the energyweighted sum rule obtained in 8) recruited from the neglect of ground state correlations, as noted at the end of section b. However, one can show ,1) that the introduction of finite range exchange forces works in the opposite direction, so that using our formalism, the energy-weighted sum rule comes out correct to first order in the amount of exchange force. Therefore, the effects of introducing forces of finite range and ground-state correlations seem to work against each other, and leaving both out probably gives better results than leaving either one out singly. This is, of course, no convincing excuse for not making better calculations, and ones with inclusion of both of these effects are now in progress. We note that the Elliott and Flowers Calculation lo) cannot be relied upon to give good results for the energy-weighted dipole sum rule, since it neglects ground-state correlations. The T = 1 dipole and T = 0 octupole states are probably the vibrations of spherical nuclei easiest to see experimentally. However, our development points out the richness of possibility of other viblational states, not yet observed. For example, a T = 1 octupole state should exist somewhere below the T = 1 dipole state in energy. Here, again, the octupole strength will be. sp~it between two states because of the effect of spin. Calculations by E. Sanderson, carries out with zero-range forces as in 8), predict energies of 18.5 and 25 MeV for these states in O TM. Another interesting possibility is the spin-flip T = 1 monopole state, suggested by Ben Mottelson and others. Tile simplest example of this would seem to be the 15 MeV state in Carbon, vchich consists of only one singlepaI~icle excitation, In heavier nuclei, a number of single-particle excitations wiii enter in. Another possibility is a giant 2/" = 1, M2 resonance, which would be ~asily exalted in inelastic particle scattering. It can be seen that a large n,,~mber of possibilities exist, and with improved experimental techniques more slates should be seen. I11 conclusion, we should like to thank the many people with whom we have had pleasant di~;cussions, including L. Castillejo, Ben Mottelson and T. H. R. Skyrme. We should also like to acknowledge receiving a preprint on work very closely related to the above by M. Barranger, who treats, in addition, quadrupole vibrations of partly-filled shells. Most of this work was carried out when
VIBRATIONS
OF SPHERICAL
NUCLEI
17
the first author (G. E. B.) was at the University of glrmllq:., ~" ' ~,~:~-tlT1, a n d wP should like to t h a n k Prof. R. E. Peierls for m u c h encouragem~nt and m a n y interesting conversations. References 1) B. L. Cohen, Piiys. Rev. 105 (1957) 1549; B. L. Cohen and A. G. Rubin, Phys. Rcv. 111 (1958) 1568 2) M. Crut, D. R. Sweetmau and N. S. Wail, Nuclear Physics 17 (1960) 655 3) R. H. Helm, Phys. Rev. 104 (1956) 1466 4) Fulbright, Lassen and Poulsen, Mat. Fys. Mcdd. Dan. Vid. Selsk. 31, No. 10 (1959); J. L. Yntema and B. Zcichman, Phys. Rev. 114 (1959) 815 5) McDaniels, Blair, Chert and Farwell, Nucler Physics 17 (1960) 614 6) A. M. Lane and E. D. Pendlebury, Nuclear Physics 15 (1960) 39 7) G. E. Brown and M. Bolstcrli, Phys. Rev. Letters 3 (1959) 472 8) Brown, Castillejo and Evans, Nuclear Physics 22 (1960) 1 9) G. E. Brown and D. J. Thouless, Pilysica (to be published) 10) J. P. Elliott and B. IK. Flowers, Proc. Roy. Soc. A 242 (1!}59) 57 11) Proc. Rehovoth Conf. on Nuclear Structure (North-Holland Publishing C~)., Amsterdan:
1958); 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
B. R. Mottelson, Varenna Lectures. 1960 Devons, Manning and Bunbury, Proc. Phys. Soc. A 68 (1955) 18 Sawada, Brucckncr, Fukuda and Brout, Phys. Rev. 108 (1957) 507 R. A. Ferrell and J. J. Quinn, Phys. Rev. 108 (1957) 57o D. Bohm and D. Pines, Phys. Rev. 92 (1953) 609 R. A. Ferrell, Phys. Rev. 107 (1957) 1631 S. Fallieros, Thesis, University of Maryland, 1959 M. Getl-Mann and F. Low, Phys. Rev. 84 (1951) 350 A. Bohr and B. R. Mottelson, Mat. Fys. 5ledd. Dan. Vid. Selsk. 27, No. 16 (1953) D. R. Inglis, Phys. Rcv. 07 (1955) 701 D. J. Thouless, Nuclear Physics 22 (1961) 78
[
Nuclear Physics 2 4 (1961) 1---17; @ North-Holland Publishi,zg, Col, Amsterdam
!
Not to be reproduced by photoprint or microfilm without written permission froth; the publisher
VIBRATIONS
OF SPHERICAL NUCLEI G. E. B R O W N
NORDITA,
Copenhagen,,
and J. A. E V A N S a n d D. J. T H O U L E S S
Mathematical Physics Deparlme~t, Universil,t ol Birmi~gham R e c e i v e d 18 N o v e m b e r 1960
A b s t r a c t : T h e p a r t i c l e - h o l e i n t e r a c t i o n is d i s c u s s e d as t h e m e c h a n i s m for p r ~ : : u c i n g vibra~tional s t a t e s in nuclei. T h e p r o c e d u r e is first i l l u s t r a t e d b y m e a n s of a s c h e m a t i c model, from w h i c h i t is s h o w n t h a t t h e u s u a l t y p e of s h e l l - m o d e l c a l c u l a t i o n m u s t b e extt:n(led ~,} i n c l u d e c,>r:el a t i o n s in t h e g r o u n d s t a t e of t h e n u c l e u s b e f c , e it c a n 1)e applied to t h e c a l c u l a t i o n of xl~>rat i o n a l s t a t e s . R e s u l t s of c a l c u l a t i o n s in ]--] c o u p l i n g , b u t w i t h z e r o - r a n g e forces, are gP.:en.
1. I n t r o d u c t i o n
Considerable experimental 1-5) and theoretical 5,6) evidence s~lggest~ !:he existence of octupo!e vibrations in a large number of nucM ?-~>s~..,~sm s'',~ spheri~cal equilibrium shape. In this note we wish to indicate, that the particle-t~o:(~ interaction produces a low-lying 3-- state possessing an extremely large octupole moment and thus provides a natural mechanism for such a state. The techniques will be outlined in general form, however, so as to be applicable to other cases. It has been suggested 7, s, 9) that the particle-hole interaction is the mechan:ism producing the dipole state (giant photonuclear resonance), and zero-range calculations showed it to be extremely effective both in tran:¢ferring dipole strength to the dipole state and in raising tI~e energy of the latter. In refs. a, 9) it was pointed out that the particle-hole interaction was generally repulsive in T = 1 states and attractive in T = 0 ones. If, therefore, a T = 1 state carrying most of the multipole moment goes up in energy, then one would expect a T = 0 one to come down. The simplest cases to consider are double closed.. shell nuclei; it was pointed out in ref. s) that the valence particles played relatively little part in the state, except in determining its width. "Ihe simplest states to consider are those that arise from liI~ear combinations of particles raised from one shell to the next, i.e., excitations of ho) in the harmonic osciUator model. Now, whereas the dipole T = 1 state describes motion of neutrons against protons and, therefore, introduces no problems of spurious centre-ofmass motion, the dipole T = 0 state represents dipole motions of neutrons and 1 April 1961
2
:
~. E. BROWN et al.
protons together and is, hence, almost completely spurious. In the Elliott and Flowers calculation m), for example, this state was removed and discarded before the secular matrix was diagonalized. The next states one might look at are magnetic quadrupole or electric octupole state.s ~lthesestates have odd parity). We prefer now to look at the latte~-, since they are easily reached by inelastic particle scattering, and there is considerable evidence for their existel~ce. In this note we shall work mainly with a schematic model similar to that of refs. ~,8), quoting later results of shell-model calculations carried out in zerorange approximation. From the work with the schematic model it will become clear that the usual type of shell-model calculation will have to be modified considerably before being applied to the calculation of vibrational states because of the large energy shifts :involved. In section 2 we shall outline the treatment by the schematic model. In section 3 we consider effects from correlations present in the ground-state of the nucleus, and in section 4 we will show the classical correspondence of our schematic model. in the sections 5 and 6 we give results of shell-model calculations for O 16 and Ca a° in zero-range approximation. 2. The S c h e m a t i c Model In ref. 8) it was pointed out that we could represent the spin and isotopic spin character of a T : 1 particle-hole wave function as
"['he particle-hole interaction consi:~ts of two types of terms as shown in fig. 1, direct terms D and exchange terms E. For a Wigner force the interaction in
tl D
E
Fig, 1, Graphs corresponding to tke direct terms D and the exchange terms E of the particle-hole interaction.
T == i states was D and in T : 4) states, D - - 4 E . Although the interaction is more likely to be something like a Serber mixture plus a small spin-exchange, this will not look much different from the Wigner force in zero-range approxima-
VIBRATIONS
OF
SPHERICAL
NUCLEI
3
tion which we can consider in formulating our schematic model and which seems to reproduce qualitative features. We see t h a t the interaction in T = 0 states is d o m i n a t e d by the exchange terms. With a zero-range Wigner force, D and E are equal, so that the interaction is - - 3 E . Going to finite range decreases D compared with E, since the exchange term contains only one m o m e n t of the potential; i.e., in an expan:don
V(rl--r~) = ~ VL(rl, r.~)PL(COS 0,2)
(1]
L
only the term with L = 3 is involved in E for an octupole stat~::. This is easy t(~ see since the particle and hole coupled to angul;~r m o m e n t u m J cai1 ~i,nnihi]ate at a point as shown in fig. 1 only through the /th term in the expansion (1). On the other hand, D depends on contributions from a large number of moments of the potential Vr., and going to finite range decreases the contribution * from large L. We shall consider mainly the exchange term, therefore. i
i
1 I !
I! Fig. 2. Particle-hole crez~tion by absorption of a }J-ray,
We note, now. that if the m o m e n t VL(r ~, r.~) were factorizable and proportional to r~r r 2L, t h e n the matrix element V . , jj, between states ii' and jj', which we assume to be d o m i n a t e d by the exchange term, would be proportioilat to D . , D j : , where D . , is the amplitude for absorption of an octup(:.le y-ray. leading to creation of a particle in state i and a hole in state i', as shown in fig. '2. Now, in fact, Vr(r x, re) is nothing like rl L rez"since V r vanishes rapidly as ttl:,~ difference Ir~--rlf increases beyond 10 -la cm or so. However. we arrive at essentially the same representation of V . , z' if we keep all factor:.; coming from angular integrals, but set all radial integrals entering into both V . . ~. and D . , equal (see ref. 7)). We shall, therefore, use V.',z" = mI ,,/r/-"D.,D~j.,
(2)
where '/: is a constant determining the strength of particle-hole interaction in the schematic model. The + sigh refers to T = 1 states, the -- sign to T ~= 0 ones, and ~ will in general have different m a g m t u d e s in tlle two cases. In t h e calculations we have carried out withoua approximating the radial integrals in this way, the calculated V~:.., have often differed quite a bit from those of the schematic model. However, it has generally been found that the V.,, jj. are large t This point has been especiMly stressed b y Ben Mottelson ~md Aage Bohr,
4
G, E, BRO'~?N et a l .
(and negati~ve for T = 0 states) between states carrying large amplitudes D,e and D~j, and that they are small between states having small transition amplitudes. This seems to be sufficient to guarantee that qualitative features will com:e out as in the schematic model. Once one makes the above assumption abot~t the V,~,,~. in the T = 1 dipole case, one obtains immediately the results of ref. 7), finding in the degenerate case (all e~ in the secular matrix of that reference equal) that the dipole state is
#D = X D.'9.'/v/~D~¢,
(3)
2 and that it is pushed up an amount ~ . , ¢rDi¢ in energy. In dealing with the octupole case, we wish to obtain a solution for the case where all e~ are not equal. Thinking of the situation using a simp!e-harmonic well, for example, one has octupole absorption to states of unperturbed energies ?/~ and 3/~. and it may be important to include the latter. We prefer now to change the notation of ref. 7), so that ~i is replaced by ei---~,, where e~ is the energy of the excited particle in the potential well repres(~nting tile average interaction of tile A-particles, an4 ~, is the initial energy of the particle. In other words, e~--e,, is the unperturbed energy of the particlehole pair ii'. The SchrSdinger equation in matri!x form now reads
=
X
(4)
it"
and using the approximation of the schematic model eq. (2) for V.,, w, this is easily solved to give
ND., ~f"' = t:. --e~te,, ,
(4.1)
where N is a normalization constanL
Tilde energy of any given eigenstate may be found by taking the expectation w~ke of tile Ham iltonian, In the dege,:,erate case (all e~--e e equal to a common wdu~ ,:) the result~:~ arc particularly ~imple, the amplitudes ~v,~,being
(4.3)
~p., = N ' D . , ,
with ---- { Z
,
(4.4)
E = ~--+~ X D~,,.
(4.5)
if'
and the energy of the vibrational state if"
VIBRATIONS OF SPHERICAL NUCLEI
5
A slight generalization of the degenerate model is necessary to deal with the octupole case, since the absorbtion can proceed to states of unperturbe4 energies ~,/go and ,~ 3/~(o. Our schematic model would predict in this case (1) ~ (11
t ~ D . , v.,
(3)
(3)
,
where the upper suffices refer to states of energy 1/1o)and 31i~o, respectively'. Simple calculations then show that the second term has quite an important effect on increasing the octupole transition probability to the collective state~ Another way of including the effect of higher transitions, as suggested by Ben Mottelson, would be to introduce an effective octupole charge. However, we believe that one should ultimately try to directly calculate the effect of the higher transitions. The experiments '~) demand a higher transition probability than that obtained by Elliott and Flowers ,o), but we believe that thi~ comes about from the effects of ground-state correlations, which, we shall now 4iscu~;:~, as well as from the 3//o) transitions. 3. E f f e c t s f r o m C o r r e l a t i o n s
The necessity of taking correlations in the ground state into account in plasr~s, oscillations of an electron gas has long be recognized (See, for example, Sawada et al. 13) or FerreI1 and Quinn la)). Their inclusion is necessary to reproduce the Bohm-Pines theory is) and to make correspondence with the classical theory.
a
b
Fig. 3, a. Absorption of a ?+-ray and subsequeilt processes, b. Interact:ion of two parLicles in tim ground-state of the nucleus, forming two particle-hole pairs, a n d subsequent -~rocesses.
The necessity of including them in nuclear vibrational states has also) been pointed out le, 17). We essentially rephrase the arguments of ref. ,7) in our schematic model. Since v e deal now with vibrational states, we consider the exchange term of fig. 1, which is of main importance here. In fig. 3 we illustrate two processes which are topologically similar. Fig. 3a represents the process considered earlier, whereas fig. 3b represents two particles interacting it~, the ground state of the nucleus, forming two particle-hole pairs, one of which is
6
G.E.
BRO'~VN et al.
subsequently annihilated in absorbtion of the 7-ray. Clearly, the most conveni:ent way of including processes of the type fig. 3b is to use F e y n m a n techniques and allow for propagation in a chain either forwards or backwards in time. To this end we write down the Gell-ManmLow wave function :ts) ~,(1, 2) -~- i f f d v s d ~ 6 S v ( 1 ,
5)Sv(2, 6)1"(5, 6)~0(5, 6)
(6)
in their notation. Setting t~ = t~ ~ t' and t 5 =-. t e := t, letting coordinate 5 refer to a particle and coordinate 6 to a hole for t <2 f , and choosing /'(5, 6) = 3(ts--t~)V(5, 6),
we find in going over to an angular-momentum t
7'"'(t ) =
(6.1) representation that
. f oo dt SF~(t , --t)SFe(t r --t)~ 2".,,jj.~,~,(t),
~ -oo
(6.2)
where
we(t) ~ (ii']~p(1, 2)) = z . , e -~E'', iOie-iq(t'-O, SFi(t'-'t) =
--i(1
0 ~e-i~'tt'-°
- - i ( i ---Oe)e i~e(t'-t),
Sw.
(t' - - t) =
i O v e ~''" ( r - o ,
(t < t'), (t > o Ct ),
(6.3)
(t < t'), (t > c),
and Z.' w~mld be just the same as the previous ~p.. if one carried out the integration over only t < t'. Carrying out the integration, we find
Z." --= --
ei__av_E
~j, V..,j,r Z# ,
(7)
wh~we 0i = 1 if state i is nnoccupied and 0~ --= 0 if it is occupied. That is, the [actor 0,.(1---0e) guarantees t h a t the state i refers to a particle, the state i' to a bo!~::. The first term in eq. (7) reduces to our previous result. Tile second term rep,~.:~ents the effect of the excitation propagating backwards (these two parts are c;dled A~ ~+~ and Ai(-~ by Ferrell and Quinn 14)). Thus, although we started out ,xith i htbelling a particle and i' a hole (this designation referring to tile ba~e vacuum, where one can unambigously talk about particles and holes because of the Sharp discontinuity in occupation number at the Fermi energy), this convention has become reversed in the backwards part. However, particles can be created only in states of energy greater t h a n the Fermi energy EF, and ,holes in states of energy less than E~. Thus, i' in the second case takes on precisely the same values as i in the first case. We can preserve the designation of particles by i and holes by i' if we interchange i and i' in the second term,
VIBRATIONS OF SPHERICAL NUCLEI
7
obtaining 1 , ZJ'i(, ~" { V " " w Z ~ : + V""~'i" ~r
e~--ei ' - E
(s)
1
%-- ~.~--E ~r ~ {Vri, ~, Z:J'+ V,,~,:.:Z:i},
where Z.' refers to the forward-going part of the excitation, and Z,'~ refers to the backward-going part. The normalization is such 14,21) (see, also, eq. (14)) that Y Iz,.:-
~ Iz,.,i 2 = 1.
ti'
ii'
(s.i)
The quantities V will be discussed for the general case later. We note ti~at if one includes only the exchange term of fig. 1, i.e., graphs of the type shown in fig. 3, then Vii"
JJ' -- --
[ : i i ' , J'j =
Ve~, j j" =
V .d. G j ' j :,
(8~Zl
which is a great simplification. In this case we can add the two equations (8), obtaining 2(e,-Ee)
z.. + z,., -
(~-~,.)~--E~y'..V.. ,,.(z.,
+ z,,~),
Of
showing that the secular equation describing only the forward going part can be modified rather trivially to give the squares of the eigenenergies E 2. In the ~chematic model one finds easily that Zii"
ND., ei--ee-- E ,
ND., Zi'~ . . . . . . . . . . . . . . . , e i _ e i _t_ E
(9.1)
where N is again the normalization contant to be determined from eq. (8.1), We see that in the vibrational case when E < e~--e~,, tile.' inclu!,don of the backwards-going terms enhances the absorption. Tile energy is given by the dispersion law * 1 =
~
2 ~¢:'D~,,(ei -- ei,) .............. -)-----:-~-, (<--~,) - - E
(9.2)
and it is seen that inclusion of the backwards terms lowers the energy still more. In fact, in the degenerate case (all t.,i--e,v equal to e) E = v'~2-2~;i; 2 -~, . %- ~. b~ . . . ii" * Tiffs r e s u l t h a s also b e e n o b t a i n e d b y M. B a r a a g e r a n d Bell MottelsotL
(9.a)
s
G, F.. BROW.~et
at.
The transition amplitude in the general case is given by -----2~ D . , ( Z . , + Ze,),
(9.4)
if phase factors are chosen so that De~ = D.,, and, in the degenerate case the transition probability is given by t~:] ~ = -~ .., D 'ii',
(9.5)
that is, it is enhanced by the inclusion of backwards-going graphs by the factor e[E. The transition probability to this one state then exhausts the energyweighted sum rule, the inclusion of the backwards-going graphs being necessary for thi~ J~'17). 4. C l a s s i c a l C o r r e s p o n d e n c e
When the energy of the vibrational state becomes small compared with the single-particle excitations, the formulae shoud go over to those of the adiabatic case, and expanding in powers of E/(ei--~e) we should regain the semiclassical description t. To make the correspondence more obvious, we go over to the notation ohi. = ei--er, to = E. Then, with expansion in powers of co/~o.,, eq, (9.2) becomes
This is to be compared with the adiabatic formula 19) K ~ + M ~ " == Ko~--3Io~2~ = 0,
(11)
where our formula (10) must be essentially K - - M ~ ~ = 0 since no collective coordinate occurs. Here K represents the restoring force, and M, the inertial parameter. In fact, to within a constant, the coefficient of in eq. (10) is just the inertial parameter of the cranking model 2o). This is easy to see. The prescription of the cranking model is that one should add to the Hamiltonian H = Z Ito(i) == ~ ( T , + Vj) J
(12)
J
describing the motion of particles in a spherical well V~, the term (~[I == ~ ~ YLO(Oi, ~o~)rS L
(12.1)
in describing L th order vibrations; this additional term effectively distorts the shape of the well. Then
l(ii'l ~ !o)!3 ii' t This was suggested b y Aage Bohr.
O.~tt,
(1'.,.2)
VIBRATIONS ('F SPHERICAL NUCLEI
9
where [0> is the uncorrelated ground, state. But 0
1 OJ i i,
(ii'] ~ YL°(O~, q~j)r:r~!O) oJ i i,
(12.3) ADii, Okii,
Substituting from eq. (12.3) back into (12.2) shows that the coefficient of oJ2 in eq. (10) is, to within trivial constants, equal to M. It is not :~urprising that our eq. (2) should lead us back to the cranking-model inertial parameter, for it would be exactly fulfilled if VL(r ~ , re) were equal to rt L r.~L and we kept only the exchange terms. In this case keeping only the L th term of the nucleon-nucleon force in the expansion (1) as a residual force between particles moving in a spherical well leads directly to the particles mo,~:mg independently in a deformed we]l. The cranking model treat:; just this case. It is also not surprising that one must include ground-state correlations to make the classical eorre. spondence, since this situation is known in the treatment of plasma oscillatio~,~s in a metal 13, i,). As the number of terms in the sum of eq. (9.2) increase, or as the stre~agth of the potential '#~"increases, E moves down until it becomes zero, and any further increase in the aove factors would cause it to become imaginary. This is connected with the collapse of the nucleus into nonspherical equilibrimn shape 2,). 5. R e a l i s t i c P o t e n t i a l s
The cMculations which are described in the next section were carried out by solving eq. (8), but it was not assumed that the potential has the ~dmple form needed to give the schema,qc model of section "3. Eq. (8) is similar to a SchrSdinger equation for a wave function with components Ziv and Z~'~, but there are some important differences. We give here a brief discussion of the general properties of eq. (8), since these properties must be understood to interpret the results of the calculations. Eq. (8) can be wri,~ten as a matrix equation
where X is a colmnn vector whose components are the numbers Z,', and Y is a column vector whose components are the numbers )~v~.The equation takes this simple form, with A and B Hermitian matrices, if corresponding components of X and Y are Z,, and 2?7, where a bar over an index denotes the tinmreversed state. The matrix B has elements equal to V,,, j,j, and the matrix A has diagonal elements equal to e ~ - - e e - l - V , , , , , , and off-diagonal elements
10
G.E.
m ~ o w N et aL
V,, is" These matrices are Hermitian because of the time-reversal invariance of the shell-model energies and of the potential. Eq. (13) shows that the energy levels are given by the eigenvalues of the nonhermitian matrix
and the peculiar properties of the solutions of this equation arise from the nonhermitian property. The first thing to be noticed about eq. (13) is that for every positive eigenvalue E there is an equal negative eigenvalue --E, since eq. (13) implies (~ ~ ) ( ~ )
-=--
( Y E _X).
( ~.~'
These negative solutions have no physical significance, but describe the same excited states as the positive solutions. If we have two eigenvectors
with eigenvalues Em and En, we get
(13.2) Unless Era* and E. are equal, we must have
X~*Xn--Ym~:Y,~ -----0,
(14)
and this replaces the usual orthogonality property of eigenvectors of a Hermitian matrix. This relation shows why the normalization of the eigenvectors given i~ eq. (8.1) is the natural one to choose in this problem. A ~ufficient condition for the eigenvalues of eq. (13) to be real is that the eig~mvalues of the Hermitian matrix
should M1 be positive. This condition implies that
(A AB) ~X,,~ = E,~(X,~'X,,--YJY,,)
(15)
must be real and greater than zero, so En must be real. If the attractive part of the interaction is too strong, this condition breaks down, and imaginary eigen-
VIBRATIONS OF SPHERICAL NUCLEI
t] ~
values, representing unstable modes of vibration, may appear. It can also be shown that, if this condition is satisfied, there is a complete set of cigenvectors of eq. (13). It follows that, for any X and Y,
gives an upper bound for the magnitude of the lowest eigenvalue of eq. (13) 21). This variational principle shows that we overestimate the excitation energy of the lowest excited state when, as in section 4, we consider only a restricted number of configurations. Energy-weighted sum rules are particularly simple in this theory. We denote by Q the column vector whose components are matrix elements Q,, of an operator, such as the octupole operator, between the occupied state i' and the unoccupied state i. The matrix element of this operator between the ground state of the system and the excited state of energy E,, in our theory is
,,,,,
(~u.tOi~0) = (X.*Y,,*) O . The energy-weighted sum of the squares of these matrix elements is
where the sum is taken over all positive energy eigenvectors. From the completeness of the set of eigenvectors, and from the symmetry between positive and negative energy eigenvectors, we can show that this equation gives
~', iL(.?f.tY~*)
E,,
= . g1( Q f - Q ) I" ,
=Qt(A--B)Q.
( ~'~
If eq. (8.2) is satisfied, the interaction drops out ot this expression, and the sum is the same as it would be in the unperturbed system. This is the case in either our schematic model or in our later calculations with zero-range forces. Consequently, neither of these is adequate to describe the enhancement of the energy-weighted sum rule through effects from exchange forces in tile T = 1 case. However, it can be shown 2~) that this enhancement is contained in the formalism if oae employs forces of finite range and demands self-consistency.
6. Calculations in ]~] Couplin~ The most important results of this work are calculations in which some ground state correlations have been included as described in the preceding sections. For completeness, however, we begin by reporting results of calculations in which such effects are not included. The procedure, in this case, is precisely the
G . E. B R O W N
t:2
el al.
same as that of the earlier paper on the dipole states *), and again the work is confined to the closed shell nuclei 01~ and Ca 4°. As before 8), we indicate the effect of a realistic exchange mixture by calculating the matrices for both an ordinary force and a Soper Mixture. if only 1 /ho transitions are considered, the three configurations p½-'dj, p!:~ld[ and p~-ld~ give rise to an octupole state in 0 ~. The eigenvalues, together with the squares of the perturbed octupole moments (denoted by ~ ) are given in table la. As expected, most of the octupole strength is resident in
TABLE
1
Calculated energies and octupole strengths in 0 ~ ]
0
Ordinary force
Soper mixture
80
8.8
67
16.0
11
16.2
19
22.9
9
21.6
14
7.7
0
i
100%
~o0%
.5 3,fi
367
74
7.7
134
58
l&8
13
12
16.0
24
22
22.6
11
14
21.5
16
20
391%
lO0%
174%
100%
w,
i
!
!
I
1
is proportiomal to the octupole transition amplitude (ph[Qs[0), where Q3 = ra Y3°(0, cp). The %%; are taken with respect to ~t~.D.. 2 and 51.igei~.D.,Z, where the sum is over unperturbed wave Lm~tions, and e., = e~--e(, the unperturbed energy. All transitions considered are between adjac,e1~t ~hetls, and the sums go only over such configurations.
the ]c~,:~'est level, the energy of which could be fitted to the experimental value siml~],y by adjusting the interaction strength Vo. However, we thought it more useful to retain the value of Vo which fitted the O le dipole states 8), The eigenvector obtained for this state agreed, almost exactly with that of Elliott and Fl(Jwers, and consequently leads to their estimate of the lifetime. Using a reasonable value for the range paramef.er b of the wavefunctions, they obtained a lifetime three time,~s too long. We shall show how this situation is improved by the introduction of ground state correlation effects. For Ca ~, the results, in table 2a, are very similar. In tiffs case, nine octupole
VIBRATIONS
OF
13
SPHERICAL N U C L E I
states can be formed from the configurations d -~ f, d -~ 2p, and 2~s-~ f. Although the value of b 3 was increased in the ratio 40/16, V o was unaltered t. Correlation effects were included for both nuclei by solving the matrix equation (13) numerically. In zero-range, the calclflation of the matrix B is TABLE
2
Calculated energies and octupole strer~gths in Ca 4° E
Ordinary force (MeV)
G
6.0 8.7 9.8 11.1 12.6 14.2 16.5 16,6 17,9
",9
2 s2
1.9 8.5 9.5 ll.O 12.6 14.1 16.5 16.6 17.9
~
%
Soper mixture
E~ 2 %
62 13 15 0
2
ii
3 1
1
2
i
100%
i
2i
E(MeV)
~2 %
6.3 9.2 10.2 11.3 12.6 14.8 16.1 16.5 17.2
43 7 35 0 2 4 7 0 1
lOO%
512 8 16 0 2 3 4 0 2
72 5 11 0 2 2 5 0 3
5.4 9.0 9.9 11.3 12.6 14.1 16.1 16.5 17.2
10o%
547%
E ~ 2 %,
109 13 39 0 2 5 8 o 1
45 9 29 0 2 5 9 0 1
177°;
See table 1 for definition of the symbols.
rather shnple. The phases of the u n p e r t u r b e d states can be chosen so that all the oct'apole m o m e n t s ~ are positive and invariant under the interchange i =,- i',
i' --~-i.
T h a t is to say D.,
== Di,
i .
It follows then, t h a t the part of the matrix element correspondang to zero total ,spin S --- 0, is invariant under the interchange of particle and hole q u a n t u m numbers. A simple argument shows that under these circumstances the S -- 1 t After these cMculations were completed, it was pointed out to us by J. Blomquist t h a t one should take the r.m.s, radius (fi)t to be proportional to A t and not b. This would involve a sub:stantial increase in Vo for (,~,,,0 b u t would not change our results quMitatively.
6. E. BROWN et al.
14
part always changes sign. Now the S = 1 terms in the interaction can arise only as a result of spin-orbit forces, so that if the latter are negelected, the relations (8.2) must be satisfied (in zero-range approximation). In the present calculations, however, spin-orbit forces are included, and therefore B differs frc,m the interaction part of A. Two important features of the results for 0 ~6 in table lb, and for Ca ~ in table 2b, are that for both types of force, the energies of the lowest states are lower than the corresponding ones in la and 2a, and that these lowest states have become very sensitive to the exchange mixt a r e . It can be shown quite generally that the lowest eigenvalue must be lower than the lowest eigenvalue of the matrix A. We now turn r.o a discussion of sum rules. Lane and Pendlebury 6) have considered the fractions of certain sum rules exhausted by the lowest octupole s t a t e s in various nuclei, including O a6 and Ca4°. First they used the sum of squares of matrix elements E D~.,. ii'
and find, for the lowest 3- state in O ~6 that ~
(17)
~ 0 . 6 4 ~ P . ~, . ii'
Now, the total sum contains contributions from both l h ~ and 3tUo configurations, so that = ~i'
D., q- E
i,',
ii"
where the upper suffices "1" and "3" refer to ltwj and 3hoJ, respectively. It turns ou~ for 0 ~ that D(a?-
ii" i b.nJ:, :~ e,
ii"
__
--
7
-3 ~ ii'
D(n~
ii" "
from (17)
~@2= 0.64(1
+~)Z D(1)°'=ii'2 . 1 3 if"
l.~ ~ ~'it'/)(1)2
(18)
if'
Retatior, (18) :is the more convenient form to use here because we have not explicitly irmluded any 3ti~,~effects in our calculations. If the results of table lb (for the 5oper Mixture) are recalculated with a force strength V0 which tits the ]cnvest energy to the experimental value of 6.13 MeV ¢, then c~2 rises from 134 % to 184 %. This latter figure is to be compared with the 213 % in (18). If one does not simply renormalize, but tries to make an estimate, of what elfect t h e 3ho~ transitions would have on 2 *, one finds that they would give This "renormalization" m a y be thought of as taking some account of the 3ho9 transitions, inch~sion of which wouM push the level down. This is in the spirit of using an effective charge for "/¢ay transitions.
V I B R A T I O N S OF S P H E R I C A L N U C L E I
15
rise to an enhancement of about 40 %, leading to ~2
___ 1 . 3 4 ×
× ~ D(1)~ir= 1 . 8 8 Y~ l) wa)~,
1.40
ii'
ii"
in agreement with results from renormalizing the interaction. It is interesting to consider the reason underlying our large ~2. In the normal shell model, in which ground state correlations are not included, the orthonormality of the eigenvectors implies that,
Z n ('>' .
=
--it'
(l:))
'
it"
where the index p goes over all perturbed stat(.. From this it is clear that. for the iowest level, < D it"
making the result (18) unobtainable. Of course, the situation w(mld be improved by including 3h~o transitions, but it turns out that these do not mix in sufficiently strongly. It is, in any case. clear that there is no possibility within the framework of the shell model with 1/i.w transitions of obtaining a correct value for the matrix element, and hence of the lifetime of the low 3- resonance ;n O ~(~. When ground state correlations are introduced, (19) is no longer sati~fied. and indeed there is no theoretical limit on the left hand side; it will b(c)m( infinite as the lowest eigenvalue drops to zero. In this case, pr )~Acd only that zero range forces are used (and, in fact, for finite-range forces if self-consistency is enforced ~.1)), X E v ' c> ~ . 2 = X E~ . ' -l-). a)~ " . P
ii'
and E., being the perturbed and unperturbed eigenvalues re+;pectively. In the present case, we obtained with E =: 6.13. the result ED
E ~ 2 = 0.65 ~ E. . . L) (1) ., '' .
(20)
ig"
Now ideally the total energy-weighted sum would be given by
X E w l )2. .
=
8 ~. L~-i v I J i ir~tl): , ,
ti"
t(
and (20) would become O. ~a E~2
--
8
X Ei,:D " 2. , iY
:
0.08 ~ E.., 1)"' _.,. ig'
Increasing this by 40 % as before, we would obtain E,~ 2 ~ O.II ~, ~F., t '~-,l) ~i f p , if'
(21)
16
G.E.
BROWN et al.
which, bearing in mind the uncertainties in this procedure, agrees quite well with the estimate of 0.18 of this sum rule given by Lane and Pendlebury e).
7. Summary In this and a preceding paper 8), the origin of vibrational states through the mixing of shell-model configurations by the particle-hole force has been discussed. The calculations carried out in these papers were only rough ones, and, in some cases, were inconsistent. For example, the enhancement of the energyweighted sum rule obtained in 8) recruited from the neglect of ground state correlations, as noted at the end of section b. However, one can show ,1) that the introduction of finite range exchange forces works in the opposite direction, so that using our formalism, the energy-weighted sum rule comes out correct to first order in the amount of exchange force. Therefore, the effects of introducing forces of finite range and ground-state correlations seem to work against each other, and leaving both out probably gives better results than leaving either one out singly. This is, of course, no convincing excuse for not making better calculations, and ones with inclusion of both of these effects are now in progress. We note that the Elliott and Flowers Calculation lo) cannot be relied upon to give good results for the energy-weighted dipole sum rule, since it neglects ground-state correlations. The T = 1 dipole and T = 0 octupole states are probably the vibrations of spherical nuclei easiest to see experimentally. However, our development points out the richness of possibility of other viblational states, not yet observed. For example, a T = 1 octupole state should exist somewhere below the T = 1 dipole state in energy. Here, again, the octupole strength will be. sp~it between two states because of the effect of spin. Calculations by E. Sanderson, carries out with zero-range forces as in 8), predict energies of 18.5 and 25 MeV for these states in O TM. Another interesting possibility is the spin-flip T = 1 monopole state, suggested by Ben Mottelson and others. Tile simplest example of this would seem to be the 15 MeV state in Carbon, vchich consists of only one singlepaI~icle excitation, In heavier nuclei, a number of single-particle excitations wiii enter in. Another possibility is a giant 2/" = 1, M2 resonance, which would be ~asily exalted in inelastic particle scattering. It can be seen that a large n,,~mber of possibilities exist, and with improved experimental techniques more slates should be seen. I11 conclusion, we should like to thank the many people with whom we have had pleasant di~;cussions, including L. Castillejo, Ben Mottelson and T. H. R. Skyrme. We should also like to acknowledge receiving a preprint on work very closely related to the above by M. Barranger, who treats, in addition, quadrupole vibrations of partly-filled shells. Most of this work was carried out when
VIBRATIONS
OF SPHERICAL
NUCLEI
17
the first author (G. E. B.) was at the University of glrmllq:., ~" ' ~,~:~-tlT1, a n d wP should like to t h a n k Prof. R. E. Peierls for m u c h encouragem~nt and m a n y interesting conversations. References 1) B. L. Cohen, Piiys. Rev. 105 (1957) 1549; B. L. Cohen and A. G. Rubin, Phys. Rcv. 111 (1958) 1568 2) M. Crut, D. R. Sweetmau and N. S. Wail, Nuclear Physics 17 (1960) 655 3) R. H. Helm, Phys. Rev. 104 (1956) 1466 4) Fulbright, Lassen and Poulsen, Mat. Fys. Mcdd. Dan. Vid. Selsk. 31, No. 10 (1959); J. L. Yntema and B. Zcichman, Phys. Rev. 114 (1959) 815 5) McDaniels, Blair, Chert and Farwell, Nucler Physics 17 (1960) 614 6) A. M. Lane and E. D. Pendlebury, Nuclear Physics 15 (1960) 39 7) G. E. Brown and M. Bolstcrli, Phys. Rev. Letters 3 (1959) 472 8) Brown, Castillejo and Evans, Nuclear Physics 22 (1960) 1 9) G. E. Brown and D. J. Thouless, Pilysica (to be published) 10) J. P. Elliott and B. IK. Flowers, Proc. Roy. Soc. A 242 (1!}59) 57 11) Proc. Rehovoth Conf. on Nuclear Structure (North-Holland Publishing C~)., Amsterdan:
1958); 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
B. R. Mottelson, Varenna Lectures. 1960 Devons, Manning and Bunbury, Proc. Phys. Soc. A 68 (1955) 18 Sawada, Brucckncr, Fukuda and Brout, Phys. Rev. 108 (1957) 507 R. A. Ferrell and J. J. Quinn, Phys. Rev. 108 (1957) 57o D. Bohm and D. Pines, Phys. Rev. 92 (1953) 609 R. A. Ferrell, Phys. Rev. 107 (1957) 1631 S. Fallieros, Thesis, University of Maryland, 1959 M. Getl-Mann and F. Low, Phys. Rev. 84 (1951) 350 A. Bohr and B. R. Mottelson, Mat. Fys. 5ledd. Dan. Vid. Selsk. 27, No. 16 (1953) D. R. Inglis, Phys. Rcv. 07 (1955) 701 D. J. Thouless, Nuclear Physics 22 (1961) 78